Group #30770576 — subgroups

Subgroups — #30770576

Order 17152.

246 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 87 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22562489	C₂	1	2	yes	maximal
2	#22562482	C₂	1	2	yes	maximal
3	#22559819	C₂	1	1	yes	maximal, characteristic
4	#22558126	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#1713	C₂	1	1	yes	maximal, characteristic
6	#16863801	C₂²	1	1	yes	characteristic
7	#16863801	C₂²	1	1	yes	characteristic
8	#16863747	—	2	1	no	—
9	#16863747	—	2	1	no	—
10	#16863223	—	2	4	no	—
11	#16863221	—	2	4	no	—
12	C₄×C₆₇⋊C₁₆	C₂²	1	2	yes	—
13	#500×C₆₇	—	2	1	no	—
14	#500×C₆₇	C₂²	1	2	yes	—
15	#494×C₆₇	—	2	1	no	—
16	C₈²×C₆₇	C₂²	1	1	yes	characteristic
17	#14524043	—	2	1	no	—
18	#14524043	—	2	1	no	—
19	C₂×C₆₇⋊C₁₆	D₄	1	4	yes	—
20	#14523977	—	2	2	no	—
21	#14523977	—	2	2	no	—
22	#14523968	—	4	2	no	—
23	#14523968	—	4	2	no	—
24	C₂×Q₁₆×C₆₇	—	4	2	no	—
25	C₄⋊Q₈×C₆₇	—	2	2	no	—
26	#107×C₆₇	—	2	2	no	—
27	#107×C₆₇	—	2	2	no	—
28	#107×C₆₇	—	4	2	no	—
29	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
30	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
31	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
32	C₂×Dic₁₃₄	—	4	1	no	—
33	C₂×Dic₁₃₄	—	4	1	no	—
34	C₄⋊Dic₆₇	—	4	1	no	—
35	C₄⋊Dic₆₇	—	4	1	no	—
36	C₄⋊Dic₆₇	—	4	1	no	—
37	C₄⋊Dic₆₇	—	4	1	no	—
38	C₆₇⋊C₁₆	Q₁₆	1	8	yes	complemented
39	C₂×Q₈×C₆₇	—	4	2	no	—
40	Q₁₆×C₆₇	—	8	4	no	—
41	C₂×C₈×C₆₇	D₈	1	2	yes	—
42	C₂×C₈×C₆₇	D₈	1	2	yes	—
43	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
44	C₄⋊C₄×C₆₇	—	4	2	no	—
45	C₄⋊C₄×C₆₇	—	4	2	no	—
46	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
47	Q₈×C₆₇	—	8	4	no	—
48	C₂×C₄×C₆₇	—	8	2	no	—
49	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
50	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
51	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
52	C₂×Dic₆₇	—	8	1	no	—
53	C₂×Dic₆₇	—	8	1	no	—
54	Dic₁₃₄	—	8	2	no	—
55	Dic₁₃₄	—	8	2	no	—
56	C₈×C₆₇	—	2	2	no	—
57	C₈×C₆₇	—	2	2	no	—
58	C₈×C₆₇	C₂×Q₁₆	1	4	yes	—
59	C₂²×C₆₇	#493	1	1	yes	characteristic
60	C₄×C₆₇	—	16	2	no	—
61	C₄×C₆₇	#494	1	2	yes	—
62	C₄×C₆₇	#509	1	2	yes	—
63	C₄×C₆₇	#510	1	2	yes	—
64	Dic₆₇	—	16	1	no	—
65	Dic₆₇	—	16	1	no	—
66	#12524	—	67	1	no	maximal, Sylow
67	C₂×C₆₇	#1713	1	1	yes	characteristic
68	C₂×C₆₇	#2250	1	1	yes	characteristic
69	C₂×C₆₇	#2251	1	1	yes	characteristic
70	#2254	—	67	2	no	—
71	#2252	—	67	2	no	—
72	#1713	D₆₇	1	1	yes	characteristic
73	#1710	—	67	1	no	—
74	#1311	—	67	1	no	—
75	C₆₇	#12524	1	1	yes	characteristic, complemented, Sylow
76	#500	—	2	1	no	—
77	#500	—	67	1	no	—
78	#500	—	67	1	no	—
79	#500	C₂×D₆₇	1	2	yes	—
80	#494	—	2	1	no	—
81	#487	—	134	1	no	—
82	#487	—	134	1	no	—
83	#367	—	134	4	no	—
84	#366	—	134	4	no	—
85	C₄×C₁₆	—	67	2	no	—
86	C₈²	C₂×D₆₇	1	1	yes	characteristic
87	C₂×Q₁₆	—	4	2	no	—
88	C₄⋊Q₈	—	2	2	no	—
89	C₄⋊Q₈	—	134	1	no	—
90	C₄⋊Q₈	—	134	1	no	—
91	C₂×C₁₆	—	67	4	no	—
92	#107	—	2	2	no	—
93	#107	—	2	2	no	—
94	#107	—	4	2	no	—
95	#107	—	134	2	no	—
96	#107	—	134	2	no	—
97	#103	—	268	2	no	—
98	#103	—	268	2	no	—
99	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
100	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
101	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
102	C₂×Q₈	—	4	2	no	—
103	C₂×Q₈	—	268	1	no	—
104	C₂×Q₈	—	268	1	no	—
105	Q₁₆	—	8	4	no	—
106	C₂×C₈	#11761613	1	2	yes	—
107	C₂×C₈	#11761613	1	2	yes	—
108	C₂×C₈	D₄×D₆₇	1	2	yes	—
109	C₄⋊C₄	—	4	2	no	—
110	C₄⋊C₄	—	4	2	no	—
111	C₄⋊C₄	—	268	1	no	—
112	C₄⋊C₄	—	268	1	no	—
113	C₄⋊C₄	—	268	1	no	—
114	C₄⋊C₄	—	268	1	no	—
115	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
116	C₁₆	—	67	8	no	—
117	Q₈	—	8	4	no	—
118	Q₈	—	536	2	no	—
119	Q₈	—	536	2	no	—
120	C₂×C₄	—	8	2	no	—
121	C₂×C₄	—	536	1	no	—
122	C₂×C₄	—	536	1	no	—
123	C₂×C₄	#14524039	1	1	yes	characteristic
124	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
125	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
126	C₈	—	2	2	no	—
127	C₈	—	2	2	no	—
128	C₈	Q₁₆×D₆₇	1	4	yes	—
129	C₂²	#16863772	1	1	yes	characteristic
130	C₄	—	16	2	no	—
131	C₄	—	1072	1	no	—
132	C₄	—	1072	1	no	—
133	C₄	#16863774	1	2	yes	—
134	C₄	#16863832	1	2	yes	—
135	C₄	#16863837	1	2	yes	—
136	C₂	#22559825	1	1	yes	characteristic
137	C₂	#22562473	1	1	yes	characteristic
138	C₂	#22562478	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22562489

C₂

yes

maximal

#22562482

C₂

yes

maximal

#22559819

C₂

yes

maximal, characteristic

#22558126

C₂

yes

maximal, characteristic

C₆₇×#1713

C₂

yes

maximal, characteristic

#16863801

C₂²

yes

characteristic

#16863801

C₂²

yes

—

—

—

—

C₄×C₆₇⋊C₁₆

C₂²

yes

—

#500×C₆₇

—

#500×C₆₇

C₂²

yes

—

#494×C₆₇

—

C₈²×C₆₇

C₂²

yes

characteristic

#14524043

—

#14524043

—

C₂×C₆₇⋊C₁₆

D₄

yes

—

—

—

—

—

C₂×Q₁₆×C₆₇

—

C₄⋊Q₈×C₆₇

—

#107×C₆₇

—

#107×C₆₇

—

#107×C₆₇

—

C₄×C₈×C₆₇

D₄

yes

characteristic

C₄×C₈×C₆₇

D₄

yes

characteristic

C₄×C₈×C₆₇

C₂³

yes

characteristic

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₆₇⋊C₁₆

Q₁₆

yes

complemented

C₂×Q₈×C₆₇

—

Q₁₆×C₆₇

—

C₂×C₈×C₆₇

D₈

yes

—

C₂×C₈×C₆₇

D₈

yes

—

C₂×C₈×C₆₇

C₂×D₄

yes

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄²×C₆₇

C₂×D₄

yes

characteristic

Q₈×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

#127

yes

characteristic

C₂×C₄×C₆₇

C₂×D₈

yes

characteristic

C₂×C₄×C₆₇

C₂×D₈

yes

characteristic

C₂×Dic₆₇

—

C₂×Dic₆₇

—

Dic₁₃₄

—

Dic₁₃₄

—

C₈×C₆₇

—

C₈×C₆₇

—

C₈×C₆₇

C₂×Q₁₆

yes

—

C₂²×C₆₇

#493

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

#494

yes

—

C₄×C₆₇

#509

yes

—

C₄×C₆₇

#510

yes

—

Dic₆₇

—

Dic₆₇

—

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

—

—

D₆₇

yes

characteristic

#1710

—

#1311

—

C₆₇

#12524

yes

characteristic, complemented, Sylow

—

—

—

C₂×D₆₇

yes

—

#494

—

#487

—

134

—

#487

—

134

—

#367

—

134

—

#366

—

134

—

C₄×C₁₆

—

C₈²

C₂×D₆₇

yes

characteristic

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

134

—

C₄⋊Q₈

—

134

—

C₂×C₁₆

—

—

—

—

—

134

—

#107

—

134

—

#103

—

268

—

#103

—

268

—

C₄×C₈

C₂²⋊D₆₇

yes

characteristic

100

C₄×C₈

C₂²⋊D₆₇

yes

characteristic

101

C₄×C₈

C₂²×D₆₇

yes

characteristic

102

C₂×Q₈

—

103

C₂×Q₈

—

268

—

104

C₂×Q₈

—

268

—

105

Q₁₆

—

106

C₂×C₈

#11761613

yes

—

107

C₂×C₈

#11761613

yes

—

108

C₂×C₈

D₄×D₆₇

yes

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

268

—

112

C₄⋊C₄

—

268

—

113

C₄⋊C₄

—

268

—

114

C₄⋊C₄

—

268

—

115

C₄²

C₂×C₂²⋊D₆₇

yes

characteristic

116

C₁₆

—

117

Q₈

—

118

Q₈

—

536

—

119

Q₈

—

536

—

120

C₂×C₄

—

121

C₂×C₄

—

536

—

122

C₂×C₄

—

536

—

123

C₂×C₄

#14524039

yes

characteristic

124

C₂×C₄

C₂×#11761613

yes

characteristic

125

C₂×C₄

C₂×#11761613

yes

characteristic

126

C₈

—

127

C₈

—

128

C₈

Q₁₆×D₆₇

yes

—

129

C₂²

#16863772

yes

characteristic

130

C₄

—

131

C₄

—

1072

—

132

C₄

—

1072

—

133

C₄

#16863774

yes

—

134

C₄

#16863832

yes

—

135

C₄

#16863837

yes

—

136

C₂

#22559825

yes

characteristic

137

C₂

#22562473

yes

characteristic

138

C₂

#22562478

yes

characteristic

	subgroup	G/H
center	C₂²	#16863772
commutator, cosocle	C₄×C₈×C₆₇	C₂³
Frattini	C₄×C₈	C₂²×D₆₇
Fitting	C₆₇×#1713	C₂
socle, prime radical	C₂²×C₆₇	#493
hypercenter	#1713	D₆₇
hypocenter	C₆₇	#12524
LCS	G ▷ C₄×C₈×C₆₇ ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1713

subgroup

G/H

center

C₂²

#16863772

commutator, cosocle

C₄×C₈×C₆₇

C₂³

Frattini

C₄×C₈

C₂²×D₆₇

Fitting

C₆₇×#1713

C₂

C₂²×C₆₇

D₆₇

C₆₇

G ▷ C₄×C₈×C₆₇ ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1713

N	H ≅ G/N	type
38. C₆₇⋊C₁₆	Q₁₆	semidirect
75. C₆₇	#12524	semidirect

H ≅ G/N

type

38. C₆₇⋊C₁₆

Q₁₆

semidirect

75. C₆₇

#12524

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#22562489	2
#22562482	2
#22559819	2
#22558126	2
C₆₇×#1713	2
#12524	67

subgroup

index

C₆₇×#1713

Subgroups — #30770576

Conjugacy classes of subgroups

Complements

Maximal subgroups